Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable Explicit (TASE) operators in a unified framework. The proposed TASE operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightforwardly. The resulting time integration methods remain the original explicit time-marching schemes, yet with nearly unconditional stability. The TASE operators can be designed to be arbitrarily high-order accurate with Richardson extrapolation such that the accuracy order of original explicit time-marching method is preserved. Theoretical analyses and stability diagrams show that the s-stages sth-order explicit Runge-Kutta (RK) methods are unconditionally stable when preconditioned by the TASE operators with order $p\le s$ and $p\le 2$. On the other hand, the sth-order RK methods preconditioned by the TASE operators with order of $p\le s$ and $p>2$ are nearly unconditionally stable. The only free parameter in TASE operators can be determined a priori based on stability arguments. Unlike classical implicit methods, the TASE methodology allows for solving non-linear problems with arbitrary order without requiring solving a nonlinear system of equations. A set of benchmark problems with strong stiffness is simulated to assess the performance of the TASE method. Numerical results suggest that the proposed framework preserves the high-order accuracy of the explicit time-marching methods with very-large time steps for all the considered cases. As an alternative to established implicit strategies, TASE method is promising for the efficient computation of stiff physical problems.

无条件稳定的隐式时间前进方法有效地解决了刚性微分方程。在这项工作中，提出了一种隐式处理刚性物理项的新颖框架。源自对流，扩散和源项（通常与反应有关）的物理和数值刚度都可以在统一框架中由一组预定义的时间精确和高度稳定的显式（TASE）运算符处理。拟议的TASE运算符充当硬性条件的前提条件，可以直接部署到任何现有的显式时间前进方法中。由此产生的时间积分方法仍然是原始的显式时间行进方案，但具有几乎无条件的稳定性。通过使用Richardson外推法，可以将TASE运算符设计为任意高阶精度，从而保留原始的显式时间前进方法的精度顺序。理论分析和稳定性图表明由TASE运算符按顺序对s阶s阶s阶显式Runge-Kutta（RK）方法是无条件稳定的$p\le s$ 和 $p\le 2$。另一方面，由TASE运算符预处理的s阶RK方法的阶为$p\le s$ 和 $p>2$几乎是无条件稳定的。TASE运算符中唯一的自由参数可以根据稳定性参数先验确定。与经典的隐式方法不同，TASE方法允许以任意顺序求解非线性问题，而无需求解非线性方程组。模拟了一组具有强刚度的基准问题，以评估TASE方法的性能。数值结果表明，所提出的框架在所有考虑的情况下都保留了非常大的时间步长的显式时间前进方法的高阶精度。作为已建立的隐式策略的替代方法，TASE方法有望有效地解决刚性物理问题。